Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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3
votes
3answers
95 views

How to show that $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function

(This is a homework problem) I am trying to show that the series $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function on $\mathbb{R}$. My idea was to show that the functions ...
0
votes
1answer
39 views

Complex number, power series

Develop $\sinh z$ in powers of $z-\pi i$ to show that $$\lim_{z\to \pi i}\frac{\sinh z}{z-\pi i}=-1$$ I know that $\sinh z=\sum_{n=1}^\infty \frac{z^{2n-1}}{(2n-1)!}$. Edit: Following the hint ...
3
votes
1answer
84 views

Is $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?

If we let $\mathbb{Q}[[x]]$ be the set of all power series with rational coefficients then can we say that $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?
2
votes
1answer
43 views

Show that the binomial series satisfies: $(1+x)f'(x)=\alpha f(x)$

If $f(x) = \sum_{n=0}^{\infty}{\alpha\choose n}x^n$, show (without assuming the results of the binomial theorem) that $$(1+x)f'(x)=\alpha f(x)$$ for $|x|<1$ I've already shown that the sum ...
0
votes
1answer
14 views

“Sum of power” for prime numbers

I use Euler–Maclaurin formula, Faulhaber's formula and Bernoulli polynomials for "sum of powers" for this type $\sum_{t=1}^nt^k$. but I don't know to find compact form when sum is taken from first ...
3
votes
1answer
261 views

Which Newton's Identities are being referred to here?

I do not understand this line of Wikipedia's page on the Basel problem: If we formally multiply out this product and collect all the $x^2$ terms (we are allowed to do so because of Newton's ...
0
votes
1answer
44 views

Power Series Expansion

How can I find the Maclaurin series for $f(x)=e^x$/$(1-x^2)$? I have tried expanding it out but I am having trouble with the algebra of it.
0
votes
0answers
35 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
4
votes
0answers
34 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
4
votes
1answer
79 views

How many $s,t,u$ satisfy: $s +2t+3u +\ldots = n$?

Given $n\in \mathbb{N}^+$, what is the possible number of combinations $s,t,u,\ldots\in\mathbb{N}$, such that: $$s +2t+3u +\ldots = n\quad?$$ Additionally, is there an efficient way to find ...
118
votes
15answers
8k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
3
votes
1answer
80 views

Power series as fractions

This is what I did: \begin{equation*} (x^3-x^6)x^6[x+x^2+x^3+..], \\ \frac{(x^3-x^6)x^6}{1-x}. \end{equation*} What mistake did I make? And, How to solve this: $1+3x^2+9x^4+27x^6+...+3^{157}x^{314}$ ...
0
votes
2answers
46 views

Transformation of a function into a power series [closed]

How can I transform the real functions $\frac{1}{1-\sin x}$ and $\frac{x}{e^x-1}$ into power series with $x_0=0$?
0
votes
1answer
19 views

Series expansions of inverse polynomials

Suppose one is given a strictly monotonous polynomial, $$f(x) = \sum_{n=0}^N a_n x^n$$ So that for a given $y$ there exists a single real $x=f^{-1}(y)$. It would be nice* to be able to calculate the ...
1
vote
0answers
24 views

Decomposing a series

When I insert the following function \begin{equation} F(X,Y)=-\frac{1}{Y^{2/3}}\sum _{m=0}^{\infty } \frac{\Gamma \left(\frac{m+2}{3}\right)}{m! \Gamma (m+1)}\left(-\frac{X^2}{2^2 ...
-2
votes
2answers
86 views
1
vote
2answers
32 views

Periodicity of trigonometric functions directly from their power series

My question is very simple yet I've gotten nowhere with it. Is there any way one can, without directly or indirectly referencing any differential equations satisfied by the circular trigonometric ...
0
votes
2answers
37 views

Finding the radius of convergence of a power series, $\sum_{n=1}^{\infty} a_n x^n$.

I have to detemernine the radius of convergence of the power series $\sum_{n=1}^{\infty} a_n x^n$, where $(a_n)_{n=0,1,2,...}$ is given by $a_n=2-\dfrac{1}{2}a_{n-1}$ with $a_0=2/3$. So far I've ...
2
votes
2answers
188 views

What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
0
votes
1answer
22 views

ODE Series Solution

For the ODE: $$\frac{dy}{dx}=2y$$ If the successive derivatives calculated are: $$y'=2y,y''=2y'=(2^2)y,y^{(3)}=(2^3)y,\ldots,y^{(n)}=(2^n)y$$ How do I find the coefficients of the following ...
2
votes
1answer
368 views

Difference between power series method and Frobenius method

There is the power series method for solving ordinary differential equations: one looks for solutions of the form $\sum c_n x^n$, and derives algebraic relations between coefficients $c_n$. Then ...
1
vote
1answer
22 views

First order approximation of $F(x)=\int_0^x f(t) dt$ in the neighbourhood of $\infty$

Let $f(x)$ continuous on the real line. Then the first order approximation of $$F(x)=\int_0^x f(t) dt$$ in the neighbourhood of $0$ is: $$F(x)=\int_0^x f(t) dt\sim 0 + x f(0), \ \ \ (x\rightarrow 0)$$ ...
-3
votes
0answers
32 views

Use power series to approximate the definite integral $\int_0^{0.5} \frac{1}{1+u^5}\,du$ [closed]

Use power series to approximate the definite integral with an error less than $0.000005$: $$\int_0^{0.5} \frac{1}{1+u^5}\,du$$ Can you please walk me through it/ explain the concept? I'm having a ...
1
vote
2answers
46 views

Radius of convergence of $\sum_{n\geq 0}a_{n}x^{n}$.

Consider a series $\sum_{n\geq 0}a_{n}x^{n}$ where $a_{0}=2/3$ and $a_{n}=2-(1/2)a_{n-1}$ for all $n$. It is assumed that $2/3\leq a_{n}\leq 5/3$ for all $n\geq 1$. My problem is about determining its ...
0
votes
1answer
382 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
0
votes
1answer
87 views

Combinatorial Power Series proof [closed]

Need help proving the following involving power series $A(x)$ and $B(x)$: If $A(x)B(x)=0$ (the power series where every coefficient is 0), then $A(x)=0$ or $B(x)=0$. AND If $(A(x))^2=(B(x))^2$, ...
1
vote
0answers
20 views

Is this statement about Abelian/Tauberian theorems true?

Suppose we have some real constants $c_n \geq 0$, and know that $$\sum_{n=0}^{\infty} c_nr^n$$ converges for all $r \in (0,1)$. Suppose that the limit $$\lim_{r \uparrow 1} (1-r)\sum_{n=0}^{\infty} ...
0
votes
1answer
64 views

Generating function of a sequence

There are n lines drawn in a plane such that no 2 lines are parallel and no 3 lines are concurrent. If the plane is then divided into an regions prove that $$a_1=2,a_2=4,a_n=a_{n-1}+n \; ...
1
vote
1answer
48 views

limit of jacobi theta 2 or simple series

I have a simple problem: I need to evaluate the limit $x\rightarrow 1$ of the Jacobi Theta function 2 $$\Theta_2(m,x)=2x^{1/4}\sum_{k=0}^\infty x^{k(k+1)}\cos((2k+1)m)$$ when $m=0$, that to say ...
1
vote
1answer
19 views

How do I find the set $U$ on which this series defines a holomorphic function?

I have just come across a question that asks me to find the set $U$ on which this series defines a holomorphic function. I have trawled through my notes but I can't find anything, any help on how I ...
0
votes
3answers
220 views

Series Solution of an ODE

The ODE below is required to help compute the coefficients of function. There isnt any information about this topic in my textbook so i am just wondering how i would go about this question? In this ...
0
votes
0answers
41 views

Fourier Expressions

In the Fourier series, what are all the ways we can express: $\displaystyle\sin\left(\frac{n\cdot\pi}2\right)$ $\displaystyle\cos(n\cdot\pi)$ I know we can express as $(-1)^{(n+1)}$, and as ...
0
votes
1answer
62 views

What are the four last numbers in the series $1^1 + 2^2 + 3^3 +\cdots+3458^{3458}$?

What are the four last numbers in $1^1 + 2^2 + 3^3 +\cdots+3458^{3458}$ Hello, I have come across this question, and I have no idea how to solve it. What do you guys think?
1
vote
2answers
36 views

Evaulate/approximate a series formula $\sum_{i=1}^{n}\left ( \frac{1}{n}\right)^i \left(\frac{n-1}{n}\right)^{n-i}$

Given a fixed $n$, we define two probabilities $p_1=\displaystyle \frac{1}{n}$ and $p_2=1-p_1 = \displaystyle \frac{n-1}{n}$. The goal is to evaluate/approximate $\displaystyle \sum_{i=1}^{n} p_1^i ...
1
vote
1answer
69 views

$f(x)=\sum_{i=0}^{\infty} (x^{2^n})/(1-x^{2^{n+1}})$. Find $f(99)$.

$f(x)=\sum_{i=0}^{\infty} (x^{2^n})/(1-x^{2^{n+1}})$. Find $f(99)$. ATTEMPT: The following series can be re-written as $f(x)=\sum_{i=0}^\infty \left(\frac{1}{1-x^{2^n}}\right) \cdot \left( ...
-2
votes
1answer
44 views

Finite power series [duplicate]

I'm a student and I'm looking for a solution for the following finite power series: $$ \sum_{n=0}^m \frac{1}{n!} x^n $$ By "solution" I meant expansion of the series and finding a closed form ...
1
vote
1answer
23 views

Applying the Frobenius method to $x^2 y'' - 2x y' - 10y = 0$

Here is the equation: $$x^2 y'' - 2x y' - 10y = 0 \tag{E}$$ We want to find, using the method of Frobenius, a solution in the neighbourhood of $0$, which is here a regular-singular point. ...
3
votes
2answers
30 views

Inverse Rule for Formal Power Series

I am just really starting to get into formal power series and understanding them. I'm particularly interested in looking at the coefficients generated by the inverse of a formal power series: ...
0
votes
1answer
33 views

Expand $(e^{2x}-1-2x)/x^5$ into Laurent Series on 0<|x|<$\infty$ and classify its singularity

I guess I'm having difficulty with this because its not in the form of a polynomial expression, which is what I've been taught. Nevertheless here's what I did: I know that the expansion for ...
4
votes
2answers
28 views

Question about radius of convergence.

I want to determine the radius of convergence of the series \begin{equation*} \sum_0^\infty \frac{f^{k}(5)}{k!}(z-5)^k, \end{equation*} where $f(z) = \frac{z^2}{e^{iz}-1}$. In the solution of ...
1
vote
1answer
24 views

The autocovariance function of ARMA(1,1)

So I am reading Brockwell and Davis introduction to Time Series analysis on page 89 where he derives the ACVF of an $ARMA(1,1)$ given by: $X_t - \phi X_{t-1}=Z_t+\theta Z_{t-1}$ with ${Z_t}$ is ...
1
vote
3answers
30 views

Complex number, entire function

Let $f(z)=\frac{(e^{cz}-1)}{z}$ if $z\neq0$ and $f(0)=c$ show that f is entire Theorem:A power series represents a analytical function inside their circle of convergence. I know I could prove ...
0
votes
1answer
35 views

Complex Series proof

Integrate the Maclaurin series for$\frac{1}{1+z}$ along a path, inside the circle of convergence, going from $z'=0$ to $z'=z$ and show that $$Log(z+1)=\sum_{i=1}^\infty (-1)^{n+1}\frac{z^n}{n}, ...
0
votes
1answer
304 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
2
votes
3answers
65 views

How do you add two series together

How do you add the series $$\frac{1}{2}\left(\sum_{n=0}^{\infty}\frac{2^{n}}{(z-3)^{n+1}} + \sum_{n=0}^{\infty}\frac{(z-3)^{n}}{4^{n+1}}\right)$$ ? is this right? $$\begin{aligned} ...
4
votes
3answers
108 views

Generating series of Catalan numbers

The Catalan numbers may be defined as follows: $C_0=1$ and $$C_{n+1}=\sum_{k=0}^n C_k C_{n-k}\, .$$ One way to compute these numbers is to introduce the generating series $f(x)=\sum_{n\geq 0} C_n ...
5
votes
4answers
622 views

Number raised to power of irrational number

What is the consequence of raising a number to the power of irrational number? Ex: $2^\pi , 5^\sqrt2$ Does this mathematically makes sense? (Are there any problems in physics world where we ...
5
votes
1answer
97 views

Power series related to Bernoulli numbers

I'm reading Tenenbaum's Introduction to analytic number theory. He defines Bernoulli polynomials as the unique sequence $B_n$ such that $B_0=1$ $\forall n\geq0, B_{n+1}'(X)=(n+1)B_n(X)$ ...
3
votes
1answer
44 views

Interesting Power Series

The series is $\sum_{n=1}^{\infty} r(n)x^n$ , where $r(n)$ is defined as the divisor function. The question is , what is the radius of convergence of the power series? Maybe it is not that interesting ...
0
votes
1answer
31 views

Convergence radius of complex power series

If $a_n\neq 0$ for all $n \geq n_0$ and $\lim|\frac{b_n}{a_n}|=1$, then $\rho(S)=\rho(T)$. Since S=$\sum a_nz^n$ and T=$\sum b_nz^n$. I tried to use the definition of convergence radius $$\limsup ...